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de Polignac n.  and  pentagonal n.

First numbers that belong to both sets:

1, 17767, 20827, 31901, 55777, 61105, 68587, 81317, 95887, 171197, 192067, 195301, 200751, 213005, 281017, 325501, 329707, 364327, 419497, 500837, 520087, 570725, 859195, 868301, 907537, 971635, 974051, 1249897, 1252637, 1271901, 1376167, 1454845, 1457801, 1600117, 1612535, 1640651, 1665847, 1678517, 1791427, 1870975, 1941997, 2195545, 2283517, 2712865, 2777801, 2810557, 2847637, 2864195, 2914157, 3289301, 3419395, 3487675, 3528967, 3640267, 3810457, 3882517, 3901847, 4102747, 4298527, 4441901, 4467751, 4719727, 4799887, 4951325, 5194051, 5283755, 5334551, 5351537, 5379907, 5540165, 5603767, 5867737, 6125651, 6174247, 6253625, 6321187, 6345845, 6797897, 6829867, 7318417, 7378177, 7485017, 7565651, 7885627, 7947655, 7968385, 7996067, 8142185, 8424165, 8653205, 9024587, 9061417, 9328807, 9418801, 9441367, 9622867, 10177735, 10248187, 10397717, 10844737, 11268251, 11367137, 11408467, 11767001, 11834317, 11868047, 11910277, 12148151, 12319501, 12345307, 12526705, 12665801, 12735637, 12770627, 13078837, 13185355, 13328051, 13372801, 13724425, 13760747, 13915651, 14089805, 14581327, 14609401, 14759585, 14806675, 14872727, 15416051, 15560651, 16076977, 16343251, 16353155, 17195801, 17348501, 18299827, 18678997, 19578847, 19807117, 20069617, 21532887, 22001435, 22789657, 23118251, 23153597, 23639365, 23974007, 24250651, 24541015, 24553151, 25570897, 25682497, 25981285, 26031251, 26219051, 26432707, 26470501, 27655507, 27810301, 28056275, 28160167, 28173167, 28998215, 29541547, 29594825, 29701525, 29741587, 30345757, 30548497, 30724751, 30765497, 31037827, 31325065, 31751501, 32194717, 32487247, 32529145, 32585051, 32655001, 33090365, 33160855, 33429401, 33827127, 34802825, 35092435, 35150501, 36144967, 36691901, 37168237, 37948835, 38615677, 39211597, 39580585, 40028751, 40277095, 42021127, 42275567, 42355237, 43123885, 43559287, 44094415, 44208347, 44404001, 45141551, 45273307, 46517857, 46668337, 47138051, 47728501, 48697957, 48954697, 49573627, 49849955, 49988407, 50474501, 50613817, 50770777, 51453745, 53154337, 53655551, 53799187, 54177145, 54466001, 55046017, 55264315, 55830251, 56417467, 56804497, 57100265, 57155807, 57545357, 58891001, 59777797, 60176167, 60996005, 61782877, 62014135, 62149235, 62768707, 63645037, 63723227, 64350475, 65297107, 65773015, 67211107, 68320127, 68421397, 69213877, 70133947, 70380325, 70709501, 70956887, 71370157, 72617167, 74275535, 74613687, 74698345, 75143587, 75292295, 76658727, 77713207, 77799605, 77950917, 79014217, 79079551, 79953251, 80677667, 81008027, 82003157, 82269551, 82603151, 82714501, 82982047, 83608001, 83944301, 84213827, 84236307, 87047077, 87390217, 89254837, 89416901, 92218001, 92406277, 93255895, 93848195, 94038127, 94895197, 95181751, 95349107, 95468737, 97586501, 99043751, 99605227, 99825367.